Image processing device and method thereof

ABSTRACT

An image processing device and a method thereof. The image processing device includes a mapper to map a two-dimensional plane of an input image into a three-dimensional vector surface, a coefficient calculator to calculate a coefficient with respect to an equation of a plane formed by a plurality of pixels mapped by the mapper, and an interpolator to interpolate by calculating a gray-level of a location to be interpolated based on the equation of the plane obtained by the coefficient calculator. When the pixels on the 2-D plane are mapped into the 3-D vector space, the image processing device according to an exemplary embodiment of the present general inventive concept prevents overshoot or undershoot which may occur at the edges or on planes, and thus displays a smooth image.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of Korean Patent Application No.2003-81522 filed on Nov. 18, 2003 in the Korean Intellectual PropertyOffice, the disclosure of which is incorporated herein by reference andin its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present general inventive concept generally relates to an imageprocessing device and a method thereof. More specifically, the presentgeneral inventive concept relates to an image processing device capableof resizing an input image without incurring image deterioration atedges, and a method thereof.

2. Description of the Related Art

An image signal input to a display carries a limited amount ofinformation. When an image, which is the collection of such imagesignals, is resized according to a size of the display or according to auser's desired size, the image on the display may be distorted. Forexample, if the input image is enlarged twice, each source data ismapped into a destination data of the enlarged image as shown in FIG. 1.Specifically, the source data A of coordinates (0, 0) is mapped to thedestination data of coordinates (0, 0), the source data B of coordinates(0, 1) is mapped to the destination data of coordinates (0, 2), thesource data C of coordinates (1, 0) is mapped to the destination data ofcoordinates (2, 0), and the source data D of coordinates (1, 1) ismapped to the destination data of coordinates (2, 2). The source data A,B, C and D represent pixel values, respectively. The destination data ofcoordinates (0, 1), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3), (2, 1), (2,3), (3, 0), (3, 1), (3, 2) and (3, 3) remain empty without being mappedto the source data, which causes distortion of the image signal.

Interpolation technology is required to address the above drawbacks. Theinterpolation re-samples the image data to determine values betweendefined pixels. That is, the interpolation creates values between thepixels when enlarging the image signal. Frequently used interpolationalgorithms include a nearest neighbor pixel interpolation, a bilinearinterpolation, and a cubic convolution interpolation.

The nearest neighbor pixel interpolation outputs values between pixelsby selecting values of the nearest neighbor pixels as shown in FIG. 2.This interpolation does not deteriorate the original image data, but mayintroduce errors of 1/√{square root over (2)} in maximum and coarseimage since output pixels are obtained from the collection of inputpixels without generating new data.

The bilinear interpolation outputs values between pixels by selecting aweighted average with respect to distances to four adjacent pixels asshown in FIG. 3. The bilinear interpolation rearranges the pixel valuesof the interpolated image based on the following equation.

[Equation 1]R=(1−u)(2−v)P _(x,y) +u(1−v)P _(x+1,y)+(1−u) vP _(x,y+1) +uvP _(X+1,y+1)

In Equation 1, P indicates a pixel value of the input image signal, andR indicates a pixel value of the interpolated image signal.

The cubic convolution interpolation outputs an interpolated pixel valuefrom the nearest 16 pixel values of the original image surrounding apoint to be rearranged, as illustrated in FIG. 4. In a case that asample data of the original image is ƒ(x_(k)), a continuous function{circumflex over (ƒ)}(x) interpolated from the sample data is expressedas the following equation.

$\begin{matrix}{{\hat{f}(x)} = {\sum\limits_{k}^{\;}\;{c_{k}{\beta\left( {x - x_{k}} \right)}}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

In Equation 2, β(x) is a basis function, c_(k) is a coefficient relatingto the pixel value ƒ(x_(k)), x indicates an interpolation point, andx_(k) and x_(k+1) indicate pixel locations at a current resolution. Thecontinuous function expressed as Equation 2 may be rearranged to a sinc(x) function for the sake of a more ideal interpolation as thefollowing equation:

$\begin{matrix}{{\hat{f}(x)} = {\sum\limits_{k}^{\;}\;{{f\left( x_{k} \right)}\sin\mspace{11mu}{{c\left( {x - x_{k}} \right)}.}}}} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack\end{matrix}$

However, it is infeasible to implement substantially because the sinc(x) function is defined in an infinite extent. Thus, the cubicconvolution interpolated continuous function is suggested instead of thesin c(x) function.

Relation between an interpolation point x to be interpolated andneighboring sampling points is s=x−x_(k), and 1−s=x_(k+1)−x, where 0≦s≦1and x_(k)≦x≦x_(k+1).

If β(x) of Equation 2 is used instead of sin c(x) in Equation 3, finiteextent is introduced. That is, a basis kernel function has a valid valuein the (−2, 2) region.

$\begin{matrix}{{\beta(x)} = \left\{ \begin{matrix}{{\left( {\alpha + 2} \right){x}^{3}} - {\left( {\alpha + 3} \right){x}^{2}} + 1} & {0 \leq {x} \leq 1} \\{{\alpha{x}^{3}} - {5\alpha{x}^{2}} + {8\alpha{x}} - {4\alpha}} & {1 \leq {x} \leq 2}\end{matrix} \right.} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

If β(x) of Equation 2 substitutes sin c(x) of Equation 3 using therelation between the interpolation point and the sampling points, thecubic convolution interpolated continuous function is expressed as thefollowing equation:

$\begin{matrix}\begin{matrix}{{\hat{f}(x)} = {{{f\left( x_{k - 1} \right)}\left\{ {\alpha\left( {s^{3} - {2s^{2}} + s} \right)} \right\}} +}} \\{{{f\left( x_{k} \right)}\left\{ {{\alpha\left( {s^{3} - s^{2}} \right)} + \left( {{2s^{3}} - {3s^{2}} + 1} \right)} \right\}} +} \\{{f\left( x_{k + 1} \right)}\left\{ {{\alpha\left( {{- s^{3}} + {2s^{2}} - s} \right)} + \left( {{{- 2}s^{3}} + {3s^{2}}} \right\} +} \right.} \\{{f\left( x_{k + 2} \right)}{\left\{ {\alpha\left( {{- s^{3}} + s^{2}} \right)} \right\}.}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

In equation 5, α is a cubic convolution interpolation coefficient.

The cubic convolution interpolation produces a smoother image than thenearest neighbor pixel interpolation and a sharper image than thebilinear interpolation. However, the cubic convolution interpolation maybring about image overshoot or undershoot. The generated overshoot orundershoot deteriorates the image quality.

SUMMARY OF THE INVENTION

The foregoing and/or other aspects and advantages of the present generalinventive concept are achieved by providing an image processing deviceincluding a mapper to map a two-dimensional plane of an input image intoa three-dimensional vector surface, a coefficient calculator tocalculate a coefficient with respect to an equation of a plane formed bya plurality of pixels mapped by the mapper, and an interpolator tointerpolate by calculating a gray-level of a location to be interpolatedbased on the equation of the plane obtained by the coefficientcalculator.

Additional aspects and advantages of the present general inventiveconcept will be set forth in part in the description which follows and,in part, will be obvious from the description, or may be learned bypractice of the general inventive concept.

The mapper may include a plane classifier to classify a surface formedby four pixels of the input image into a plane shape orthogonal to theZ-axis, a plane shape inclined to one direction, and a shape where thetwo planes make contact, and a direction searcher to search an edgedirection of the surface formed by the four pixels when the surfaceformed by the four pixels, which is classified by the plane classifier,is where the two planes make contact. The image is mapped based on theclassified shape of the plane classifier and the edge direction searchedby the direction searcher.

The coefficient calculator calculates the coefficient based on threecoordinates (x₀, y₀, z₀), (x₁, y₁, z₁), (x₂, y₂, z₂) on a same plane inaccordance with the following equation of the plane:ax+by+cz=1

where a, b, and c are coefficients, and

$\begin{matrix}{{a = \frac{D_{x}}{D}},{b = \frac{D_{y}}{D}},{c = \frac{D_{z}}{D}}} \\{D = {\begin{pmatrix}x_{0} & y_{0} & z_{0} \\x_{1} & y_{1} & z_{1} \\x_{2} & y_{2} & z_{2}\end{pmatrix}.}}\end{matrix}$

The interpolator acquires the gray-level based on the coefficients a, b,and c acquired by the coefficient calculator in accordance with thefollowing equation:

$z = {{{- \frac{a}{c}}x} - {\frac{b}{c}y} + {\frac{1}{c}.}}$

The image processing device further comprises a line memory to form animage laid in a two-dimensional plane by collecting image signalsdelayed in horizontal and vertical directions.

The image processing device provides an image processing methodcomprising the operations of mapping a two-dimensional plane of an inputimage into a three-dimensional vector surface, calculating a coefficientwith respect to an equation of a plane formed by a plurality of pixelsmapped at the mapping operation, and interpolating by calculating agray-level of a location to be interpolated based on the equation of theplane.

The image processing method may further comprise the operations ofclassifying a surface formed by four pixels of the input image into aplane shape orthogonal to the Z-axis, a plane shape inclined to onedirection, and a shape where the two planes contact, and searching anedge direction of the surface formed by the four pixels when the surfaceformed by the four pixels, which is classified at the searchingoperation, is where the two planes make contact. The mapping operationmaps the image based on the classified shape of the classifyingoperation and the edge direction of the classifying operation.

The classifying a coefficient operation calculates the coefficient basedon three coordinates (x₀, y₀, z₀), (x₁, y₁, z₁), (x₂, y₂, z₂) on a sameplane in accordance with the following equation of the plane:ax+by+cz=1

where a, b, and c are coefficients, and

$\begin{matrix}{{a = \frac{D_{x}}{D}},{b = \frac{D_{y}}{D}},{c = \frac{D_{z}}{D}}} \\{D = {\begin{pmatrix}x_{0} & y_{0} & z_{0} \\x_{1} & y_{1} & z_{1} \\x_{2} & y_{2} & z_{2}\end{pmatrix}.}}\end{matrix}$

The interpolating operation acquires the gray-level based on thecoefficients a, b, and c acquired at the calculating coefficientoperation in accordance with the following equation:

$z = {{{- \frac{a}{c}}x} - {\frac{b}{c}y} + {\frac{1}{c}.}}$

A computer-readable recording medium containing code providing an imageprocessing method comprises the operations of mapping a two-dimensionalplane of an input image into a three-dimensional vector surface,calculating a coefficient with respect to an equation of a plane formedby a plurality of pixels mapped at the mapping operation, andinterpolating by calculating a gray-level of a location to beinterpolated based on the equation of the plane.

Therefore, the image processing device can resize the image withsimplicity without causing image overshoot or undershoot.

BRIEF DESCRIPTION OF THE DRAWINGS

These and/or other aspects and advantages of the general inventiveconcept will become apparent and more readily appreciated from thefollowing description of exemplary embodiments, taken in conjunctionwith the accompanying drawing figures of which:

FIG. 1 is a diagram illustrating a source data mapping when enlarging animage;

FIG. 2 is a diagram illustrating a nearest neighbor pixel interpolation;

FIG. 3 is a diagram illustrating a bilinear interpolation;

FIG. 4 is a diagram illustrating a cubic convolution interpolation;

FIG. 5 is a block diagram of an image processing device according to anexemplary embodiment of the present general inventive concept;

FIG. 6 is a flowchart of image processing operation according to anexemplary embodiment of the present general inventive concept;

FIG. 7 is a diagram illustrating an edge direction and a gradient whenmapping a 2-dimensional plane into a 3-dimensional space;

FIG. 8 is a diagram illustrating when two planes are in contact whenmapping the 2-D plane into the 3-D space;

FIG. 9 is a diagram illustrating a search of a edge direction;

FIG. 10 is a diagram illustrating where a coefficient with respect tothe equation of the plane and a gray-level are calculated;

FIG. 11 is an enlarged diagram of FIG. 8; and

FIG. 12 is a diagram illustrating an example of a processed imageaccording to an exemplary embodiment of the present general inventiveconcept.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Reference will now be made in detail to the embodiments of the presentgeneral inventive concept, examples of which are illustrated in theaccompanying drawings, wherein like reference numerals refer to the likeelements throughout. The embodiments are described below in order toexplain the present general inventive concept by referring to thedrawings.

FIG. 5 is a block diagram of an image processing device according to anexemplary embodiment of the present general inventive concept. Referringnow to FIG. 5, the image processing device includes a line memory 110, amapper 120, a coefficient calculator 130, an interpolator 140, and ascaler 150. The mapper 120 includes a plane classifier 123 and adirection searcher 125.

FIG. 6 is a flowchart of image processing operations of the imageprocessing device of FIG. 5, which is described in detail below.

The line memory 110 makes up a horizontal row by delaying input imagesignals in sequence, and makes up a column of a matrix form by delayingthe lined image signals in a vertical direction in sequence. Hence, animage is formed from the collection of the image signals delayed in thehorizontal and vertical directions. The image formed by the line memory110 is laid on a 2-dimensional (2-D) plane.

The plane classifier 123 of the mapper 120 classifies the 2-D plane ofthe input image into a plane shape which is orthogonal to a Z-axis withrespect to a surface formed by four pixels, a plane shape inclined toone direction, and a shape where two planes are in contact, at operationS601. Referring to FIG. 7, when the 2-D plane of the image is mapped toa 3-D space, such a plane classification has an edge direction accordingto a level of intensity and a gradient which is orthogonal to the edgedirection. Advantageously, the four pixels of the input image are set tofour neighboring pixels for the sake of accurate shape classification.The four pixels are not limited to this example and it is possible touse pixels at different locations.

If the surface formed by the four pixels, which is classified by theplane classifier 123, is where the two planes contact at operation S603,the direction searcher 125 of the mapper 120 searches the edge directionof the surface formed by the four pixels at operation S605.Specifically, if four pixels on the 2-D plane is mapped into the 3-Dspace, the four pixels are mapped to a plane shape orthogonal to theZ-axis, a plane shape inclined to one direction, or a shape where twoplanes are in contact as shown in FIG. 8. Referring to FIG. 8, numbersat each vertex of the 2-D plane indicate an intensity level of eachpixel. If the four pixels on the 2-D plane are mapped to the shape wherethe two planes make contact on the 3-D space, the edge direction mayvary according to the shape. The direction searcher 125 obtains the edgedirection by connecting adjacent pixels having a level similar to theinput pixel. It is advantageous that the direction searcher 125 searcheswith respect to one direction a difference of pixel values on a lineconnecting the adjacent pixels of the level similar to the input pixel,and then searches with respect to the other direction in the samemanner.

The mapper 120 maps the image based on the shape classified by the planeclassifier 123 and the edge direction searched by the direction searcher125 at operation S607. In detail, after the plane classifier 123classifies the surface shape and the direction searcher 125 searches theedge direction, the mapper 125 maps the individual pixels of the 2-Dplane into the 3-D space based on the surface shape and the edgedirection.

The coefficient calculator 130 calculates a coefficient with respect toan equation of the surface formed by the individual pixels mapped intothe 3-D space by the mapper 120 at operation S609. Referring to FIG. 10,in a case that a normal vector with respect to the plane formed by eachof the mapped pixels is (a, b, c), the equation of the plane isexpressed as below.

[Equation 6]ax+by+cz=1

If a certain three points (x₀, y₀, z₀), (x₁, y₁, z₁), (x₂, y₂, z₂) aresubstituted into Equation 6, the following equation is introduced.

[Equation 7]ax ₀ +by ₀ +cz ₀=1ax ₁ +by ₁ +cz ₁=1ax ₂ +by ₂ +cz ₂=1

Each coefficient a, b, c is calculated according to Cramer's rule inaccordance with Equation 7, which is:

$\begin{matrix}{{a = \frac{D_{x}}{D}},{b = \frac{D_{y}}{D}},{c = \frac{D_{z}}{D}},{and}} \\{D = \begin{pmatrix}x_{0} & y_{0} & z_{0} \\x_{1} & y_{1} & z_{1} \\x_{2} & y_{2} & z_{2}\end{pmatrix}}\end{matrix}$

The plane classifier 123 of the mapper 120 detects a singular valuewhich makes D=0, and the equation of the surface is acquired by applyingother equations, for example, cz=1, ax+cz=1, or by+bz=1.

When the coefficients a, b, c are acquired by the coefficient calculator130, the interpolator 140 interpolates a gray-level of the mapped pixelsby calculating the gray-level of the location to be interpolated fromthe equations of the planes applying the individual coefficients atoperation S611. In FIG. 10, since the z value indicates the gray-levelwith respect to the equation of the plane in the mapped 3-D space, theinterpolator 140 acquires the gray-level based on the followingequation.

$\begin{matrix}{z = {{{- \frac{a}{c}}x} - {\frac{b}{c}y} + \frac{1}{c}}} & \left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack\end{matrix}$

If the surface mapped into the 3-D space is enlarged by the scaler 150as shown in FIG. 11, the interpolator 140 interpolates the image byfilling an empty location with the gray-level of the locationcorresponding to the pixel to be interpolated.

When the pixels on the 2-D plane are mapped into the 3-D vector space,the image processing device according to an exemplary embodiment of thepresent general inventive concept prevents overshoot or undershoot whichmay occur at the edges or on planes, and thus displays the smooth imageas shown in FIG. 12.

The present general inventive concept can be realized as a method, anapparatus, and a system. When the present general inventive concept ismanifested in computer software, components of the present generalinventive concept may be replaced with code segments that are necessaryto perform the required action. Programs or code segments may be storedin media readable by a processor, and transmitted as computer data thatis combined with carrier waves via a transmission media or acommunication network.

The media readable by a processor include anything that can store andtransmit information, such as, electronic circuits, semiconductor memorydevices, ROM, flash memory, EEPROM, floppy discs, optical discs, harddiscs, optical fiber, radio frequency (RF) networks, etc. The computerdata also includes any data that can be transmitted via an electricnetwork channel, optical fiber, air, electromagnetic field, RF network,etc.

Although a few embodiments of the present general inventive concept havebeen shown and described, it will be appreciated by those skilled in theart that changes may be made in these embodiments without departing fromthe principles and spirit of the general inventive concept, the scope ofwhich is defined in the appended claims and their equivalents.

1. An image processing device comprising: a mapper to map atwo-dimensional plane of an input image into a three-dimensional vectorsurface; a coefficient calculator to calculate a coefficient withrespect to an equation of a plane formed by a plurality of pixels mappedby the mapper; and an interpolator to interpolate by calculating agray-level of a location to be interpolated based on the equation of theplane obtained by the coefficient calculator.
 2. The image processingdevice of claim 1, wherein the mapper comprises: a plane classifier toclassify a surface formed by four pixels of the input image into a planeshape orthogonal to the Z-axis, a plane shape inclined to one direction,and a shape where the two planes contact; and a direction searcher tosearch an edge direction of the surface formed by the four pixels whenthe surface formed by the four pixels, which is classified by the planeclassifier, is where the two planes contact, and the image is mappedbased on the classified shape of the plane classifier and the edgedirection searched by the direction searcher.
 3. The image processingdevice of claim 2, wherein the coefficient calculator calculates thecoefficient based on three coordinates (x₀, y₀, z₀), (x₁, y₁, z₁), (x₂,y₂, z₂) on a same plane in accordance with the following equation of theplane:ax+by+cz=1 where a, b, and c are coefficients, and${a = \frac{D_{x}}{D}},{b = \frac{D_{y}}{D}},{c = \frac{D_{z}}{D}}$$D = {\begin{pmatrix}x_{0} & y_{0} & z_{0} \\x_{1} & y_{1} & z_{1} \\x_{2} & y_{2} & z_{2}\end{pmatrix}.}$
 4. The image processing device of claim 3, wherein theinterpolator acquires the gray-level based on the coefficients a, b, andc acquired by the coefficient calculator in accordance with thefollowing equation:$z = {{{- \frac{a}{c}}x} - {\frac{b}{c}y} + {\frac{1}{c}.}}$
 5. Theimage processing device of claim 1, further comprising a line memory toform an image laid in a two-dimensional plane by collecting imagesignals delayed in horizontal and vertical directions.
 6. An imageprocessing method comprising the operations of: mapping atwo-dimensional plane of an input image into a three-dimensional vectorsurface; calculating a coefficient with respect to an equation of aplane formed by a plurality of pixels mapped in the mapping operation;and interpolating by calculating a gray-level of a location to beinterpolated based on the equation of the plane.
 7. The image processingmethod of claim 6, further comprising the operations of: classifying asurface formed by four pixels of the input image into a plane shapeorthogonal to the Z-axis, a plane shape inclined to one direction, and ashape where the two planes make contact; and searching an edge directionof the surface formed by the four pixels when the surface formed by thefour pixels, which is classified at the classifying operation, is wherethe two planes make contact, and the mapping operation maps the imagebased on the classified shape of the classifying operation and the edgedirection of the searching an edge direction operation.
 8. The imageprocessing method of claim 7, wherein the calculating a coefficientoperation calculates the coefficient based on three coordinates (x₀, y₀,z₀), (x₁, y₁, z₁), (x₂, y₂, z₂) on a same plane in accordance with thefollowing equation of the plane:ax+by+cz=1 where a, b, and c are coefficients, and${a = \frac{D_{x}}{D}},{b = \frac{D_{y}}{D}},{c = \frac{D_{z}}{D}}$$D = {\begin{pmatrix}x_{0} & y_{0} & z_{0} \\x_{1} & y_{1} & z_{1} \\x_{2} & y_{2} & z_{2}\end{pmatrix}.}$
 9. The image processing method of claim 8, wherein theinterpolating operation acquires the gray-level based on thecoefficients a, b, and c acquired at the calculating a coefficientoperation in accordance with the following equation:$z = {{{- \frac{a}{c}}x} - {\frac{b}{c}y} + {\frac{1}{c}.}}$
 10. Acomputer-readable recording medium containing code providing an imageprocessing method comprising the operations of: mapping atwo-dimensional plane of an input image into a three-dimensional vectorsurface; calculating a coefficient with respect to an equation of aplane formed by a plurality of pixels mapped at the mapping operation;and interpolating by calculating a gray-level of a location to beinterpolated based on the equation of the plane.
 11. Thecomputer-readable recording medium of claim 10, wherein the methodfurther comprises the operations of: classifying a surface formed byfour pixels of the input image into a plane shape orthogonal to theZ-axis, a plane shape inclined to one direction, and a shape where thetwo planes make contact; and searching an edge direction of the surfaceformed by the four pixels when the surface formed by the four pixels,which is classified at the classifying operation, is where the twoplanes make contact, and the mapping operation maps the image based onthe classified shape of the classifying operation and the edge directionof the searching an edge direction operation.
 12. The computer-readablerecording medium of claim 11, wherein the calculating a coefficientoperation calculates the coefficient based on three coordinates (x₀, y₀,z₀), (x₁, y₁, z₁), (x₂, y₂, z₂) on a same plane in accordance with thefollowing equation of the plane:ax+by+cz=1 where a, b, and c are coefficients, and${a = \frac{D_{x}}{D}},{b = \frac{D_{y}}{D}},{c = \frac{D_{z}}{D}}$$D = {\begin{pmatrix}x_{0} & y_{0} & z_{0} \\x_{1} & y_{1} & z_{1} \\x_{2} & y_{2} & z_{2}\end{pmatrix}.}$
 13. The computer-readable recording medium of claim 12,wherein the interpolating operation acquires the gray-level based on thecoefficients a, b, and c acquired at the calculating a coefficientoperation in accordance with the following equation:$z = {{{- \frac{a}{c}}x} - {\frac{b}{c}y} + {\frac{1}{c}.}}$
 14. Animage processing method, comprising: classifying a two-dimensional planeto be mapped formed by a plurality of pixels of an input image as one ofa plane shape which is orthogonal to a Z-axis with respect to a surfaceformed by the plurality of pixels, a plane shape inclined to onedirection, and a shape where two planes intersect; determining an edgedirection of the classified two-dimensional plane shape according to alevel of intensity of the plurality of pixels of the classifiedtwo-dimensional plane and a gradient which is orthogonal to the edgedirection; mapping the input image as a function of the classifiedtwo-dimensional shape and the determined edge direction into athree-dimensional vector surface; calculating a coefficient with respectto an equation of a plane formed by the mapped plurality of pixels; andinterpolating by calculating a gray-level of a location to beinterpolated based on the equation of the plane.
 15. The method of claim14, wherein the plurality of pixels is four pixels.